Let the region R be the area enclosed by the function f(x) = 2 ln (x) and g(x) = 4x – 5.Find the volume of the solid generated when the region R is revolved about the line y = -7. You may use a calculator and round to the nearest %3D thousandth. -1.5 -1 -0.5 0.5 1.5 -8 -10 -12

Q: A container can be modeled by revolving the graph of       about the x-axis, where x and y are…

A: Given function  here R(x) = y(x)

Q: b Explain why h(x)dx does not represent the area between the graph of y=h(x) and the x •Sh(x)dx a…

A: We can explain why the integral does not represent the shaded region of the area of the region .

Q: Find the area of the region bounded by the graphs of the given equations. y=2x-x, y= - 5x The area…

A: The given curves are y=2x-x2, y=-5x. Calculation: Let y1=2x-x2 and y2=-5x. Compare y2 with y=mx+c.…

Q: Find the area of the region below y = x² + x – 4 and above y = 6 for 3 < x < 6.

A: • First, we need to sketch the given parabola and identify the region whose area is to be found.…

Q: Let the region R be the area enclosed by the function f(x) = eª – 2, thel

A: Given, the function                     f(x)=ex-2 The horizontal line  y=-4, and the vertical lines…

Q: Let the region R be the area enclosed by the function f(x) = 2x³ , the horizontal line y = –3 and…

A: Given the region R be the area bounded by fx=2x13,y=-3,x=0,x=3. Visualize the solid: Each…

Q: Find the area between the curves. x = -5, x=2, y = 3x, y = x² - 4

A:

Q: b. Write an integral to find the area under the curve.

A:

Q: Let the region R be the area enclosed by the function f (x) = x³ – 1, the horizontal = 0 and x = 2.…

A: Given: The function: f(x) = x3 - 1 The horizontal line y = -4 The vertical lines are x = 0 and x = 2

Q: Consider the following equations. F(y) = y2 g(y) = y + 2 Sketch and shade the region bounded by the…

A: Use curve and line to find the area by using integration.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.